Commutators, Spectral Trace Identities, and Universal Estimates for Eigenvalues

Using simple commutator relations, we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self-adjoint operator acting in a Hilbert space. Applications involve abstract universal estimates for the eigenvalue gaps. As particular examples, we present simple proofs of the classical universal estimates for eigenvalues of the Dirichlet Laplacian (Payne-Polya-Weinberger, Hile-Protter, etc.), as well as of some known and new results for other differential operators and systems. We also suggest an extension of the methods to the case of non-self-adjoint operators.Comment: 21 pages; revised version: minor misprints correcte

Workplace Violence Against Government Employees, 1994-2011

[Excerpt] The higher rates of workplace violence in the government were partly due to the high rates of workplace violence attributed to law enforcement and security employees (figure 2). The rate of workplace violence for law enforcement and security employees was a high of 672.3 per 1,000 in 1994, declining to 109.3 in 2011. These law enforcement and security occupations accounted for over half of the violence committed against government workers and were concentrated most heavily in state, county, and local government. The estimates of nonfatal violent victimization in the workplace against government employees are based on data from the Bureau of Justice Statistics’ (BJS) National Crime Victimization Survey (NCVS), which collects information on nonfatal crimes against persons age 12 or older, reported and not reported to the police, from a nationally representative sample of U.S. households. In this report, nonfatal workplace violence includes rape or sexual assault, robbery, aggravated assault (serious violent offenses), and simple assault against employed persons age 16 or older that occurred while at work or on duty. Information on workplace homicide in this report was obtained from the Bureau of Labor Statistics’ (BLS) Census of Fatal Occupational Injuries (CFOI) (see Methodology). Workplace homicide includes the homicide of employed victims age 16 or older who were killed while at work or on duty and excludes death by accident. Trend estimates of nonfatal workplace violence are based on 2-year rolling averages centered on the most recent year. For example, estimates reported for 2011 represent the average estimates for 2010 and 2011. For some tables in this report, the focus is on the single 10-year aggregate period from 2002 through 2011. These approaches increase the reliability and stability of estimates, which facilitates comparisons over time and between subgroups. Trend estimates of workplace homicide are based on a single most recent year estimates. For example, estimates of workplace homicide for 2011 represent the estimate for 2011 only

Losing the Battle: The Challenges of Military Suicide

This report, by Dr. Margaret Harrell, CNAS Senior Fellow and Director of the Joining Forces Initiative, and Nancy Berglass, CNAS Non-Resident Senior Fellow, suggests that the health of the all-volunteer force is dependent on our nation's ability to take care of its service members and veterans.According to the report, "Suicide among service members and veterans challenges the health of America's all-volunteer force." From 2005 to 2010, service members took their own lives at a rate of approximately one every 36 hours. This tragic phenomenon reached new extremes when the Army reported a record-high number of suicides in July 2011 with the deaths of 33 active and reserve component service members reported as suicides. Additionally, the Department of Veterans Affairs estimates 18 veterans die by suicide each day. Yet the true number of veterans who die by suicide, as Harrell and Berglass point out, is unknown. As more American troops return home from war, this issue will require increasingly urgent attention. Harrell and Berglass present a number of concrete policy recommendations that will help reduce the number of service member and veteran suicides, including establishing an Army unit cohesion period; removing the congressional restriction on unit leaders discussing personally owned weapons with service members; and increasing coordination between the Department of Defense (DOD), the Department of Veterans Affairs (VA) and the Department of Health and Human Services (HHS) to improve the analysis of veteran suicide data. Despite the efforts of the DOD and the VA to address military suicide, obstacles remain, and policymakers must bring a renewed urgency to their efforts if America is to both honor the sacrifices made by the all-volunteer force and protect its future health and ability to defend the nation

On the placement of an obstacle so as to optimize the Dirichlet heat trace

We prove that among all doubly connected domains of $\R^n$ bounded by two spheres of given radii, $Z(t)$, the trace of the heat kernel with Dirichlet boundary conditions, achieves its minimum when the spheres are concentric (i.e., for the spherical shell). The supremum is attained when the interior sphere is in contact with the outer sphere.This is shown to be a special case of a more general theorem characterizing the optimal placement of a spherical obstacle inside a convex domain so as to maximize or minimize the trace of the Dirichlet heat kernel. In this case the minimizing position of the center of the obstacle belongs to the "heart" of the domain, while the maximizing situation occurs either in the interior of the heart or at a point where the obstacle is in contact with the outer boundary. Similar statements hold for the optimal positions of the obstaclefor any spectral property that can be obtained as a positivity-preserving or positivity-reversing transform of $Z(t)$,including the spectral zeta function and, through it, the regularized determinant.Comment: in SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 201